The dynamics of time-varying threshold learning

A mathematical theory of the dynamics of a class of trainable signal detectors is described. Among the constructs yielded by the theory are learning curves and variance curves. A learning curve is a curve of correct-decision probability versus training length. A variance curve is a curve of the variance of correct-decision probability versus training length. The class of trainable signal detectors to which the theory is applicable consists of all those in which the training procedure (a) raises the threshold in response to a false alarm, lowers the threshold in response to a false rest, and keeps the threshold unchanged in response to a correct decision, and (b) adjusts the size of the threshold increment by an amount that depends only on the trial number, and such that the threshold can eventually reach any real number.The “learning shape theorem” is derived. By this theorem the learning curves of all members of the above class of trainable detectors are either monotonic or single peaked.The theory is based on an approximation of the difference equation of the Markov chain of the threshold's motion by a differential equation.The theory provides an over-all view of the trade-offs between performance (the learning curves) and performance variability (the variance curves) as functions of the training length of time-varying training procedures.Several illustrative numerical examples are given